12
$\begingroup$

According to Stoke's law, the retarding force acting on a body falling in a viscous medium is given by $$F=kηrv$$ where $k=6π$.

As far as I know, the $6π$ factor is determined experimentally. In that case, how is writing exactly $6π$ correct since we obviously cannot experimentally determine the value of the constant with infinite precision?

$\endgroup$
28
$\begingroup$

It is not determined experimentally, it is an analytical result. It is verified experimentally.

As @Mick described it is possible to derive the velocity and pressure field of a flow around a sphere in the Stokes flow limit for small Reynolds numbers from the Navier-Stokes equations if the flow is further assumed to be incompressible and irrotational.

Once the flow field is determined, the stress at the surface of the sphere can be evaluated: $$\left.\boldsymbol{\sigma}\right|_w = \left[p\boldsymbol{I}-\mu\boldsymbol{\nabla}\boldsymbol{v}\right]_w$$ from which follows the drag force as: $$\left.\boldsymbol{F}\right|_w = \int_\boldsymbol{A}\left.\boldsymbol{\sigma}\right|_w\cdot d\boldsymbol{A}$$

From this it follows that the normal contribution of the drag force (form drag) is $2\pi\mu R u_\infty$, while the tangential contribution (friction drag) of the drag force is $4\pi\mu R u_\infty$, where $u_\infty$ is the free-stream velocity measured far from the sphere. The combined effect of these contributions is evaluated as $6\pi\mu R u_\infty$ or the total drag force.

This result is also found by evaluating the kinetic force by equating the rate of doing work on the sphere (force times velocity) to the rate of viscous dissipation within the fluid. This shows nicely there are often many roads to the same answer in science and engineering.

For details i suggest you look at the Chapter 2.6 and 4.2 from Transport Phenomena by Bird, Steward & Lightfoot.

$\endgroup$
1
$\begingroup$

If you have read that the coefficient is determined experimentally, then you would also have read that this applies to spherical objects with very small Reynolds numbers in a viscous fluid - Stokes' law is derived by solving the Stokes flow limit for small Reynolds numbers of the Navier–Stokes equations.

We cannot determine the value of any constant with infinite precision but we can often determine them to a level of precision where the effect of the uncertainty becomes negligible.

$\endgroup$
  • 4
    $\begingroup$ If a constant is determined (to some level of precision) experimentally, then writing it as $6\pi$, regardless of whether that's within your expected margins of error, is careless and wrong. For a concrete example of how this can go, see the history of the fine structure constant, where, as more and more accurate measurements became available, people invented more and more ridiculous expressions that it could be to make it fit within the experimental results. See this lecture by Feynman at about 19:45. $\endgroup$ – Arthur Dec 11 '17 at 13:48
  • 3
    $\begingroup$ @Arthur There is an intermediate possibility: $k$ could be constrained to be $\kappa \cdot n\pi$, with $\kappa$ an empirical constant, but $n\pi$ coming from the mathematical form of the force law. This is the case for Coulomb's law in SI units, $F = \frac{1}{4\pi\epsilon_0} \frac{q_1q_2}{r^2}$ where $\epsilon_0$ is empirical but the factor of $4\pi$ comes from taking a surface integral over a (topological) sphere. $\endgroup$ – zwol Dec 11 '17 at 18:30

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.